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Classification of Wear State for a Positive Displacement Pump Using Deep Machine Learning^{ †}

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## Abstract

**:**

## 1. Introduction

- ✓
- Diagnostic systems for hydraulics (components) based on the developed model of the diagnosed system

- ✓
- Diagnostic systems for hydraulic systems (components) based on signal analysis

- ✓
- Diagnostic systems for hydraulics (components) using the so-called intelligent fault identification

- -
- The application of the minimum redundancy maximum relevance (MRMR) algorithm in ranking the features determined from the measured signals;
- -
- Designing a neural classifier of the pump wear state based on the ranked features of the measured signals;
- -
- Preparing and carrying out a test experiment to naturally obtain the wear and tear of the pump components based on the lengthy operation of the pump at a lower oil grade;
- -
- The use of signals measured across the entire operating range of the pump (i.e., when in stationary and non-stationary operating conditions) to assess the wear state of the tested pump.

## 2. Object of the Study

## 3. Course of the Study

- A static pressure transducer;
- A dynamic pressure transducer;
- A pump body vibration acceleration transducer.

- The selection and calculation of appropriate signal features;
- The ranking of the calculated features with regard to the information they contained;
- Designing the structure of a neural network in a classifier system;
- Evaluating the effectiveness of the designed network in the classification of the wear state of the studied pump.

#### Selection of Classification System Features

## 4. Significance Ranking of Signal Features

#### Application of the Minimum Redundancy Maximum Relevance (MRMR) Algorithm in Ranking Calculated Features of Measured Signals

_{S}) and simultaneously minimized redundancy (W

_{S}). The algorithm relied on the pairwise computed mutual information (I(x, z)) of features and the mutual information of features and pump condition (I(x, y)).

_{s}—applicability of features from set S;

_{s}—redundancy of features from set S;

_{X}) (the quotient of a feature’s applicability to its redundancy). Feature classification rested upon the selection of those characterized by the largest MIQ

_{X}coefficient values (practically, larger than the assumed cut-off value).

_{X}coefficient values greater than 0.05 and the next 2 highest-ranked coefficients) were chosen to evaluate the wear state of the pump. The features that satisfied the foregoing condition are listed in Table 2.

_{X}coefficient values greater than 0.05) were chosen for the evaluation of the pump’s wear state. The features that satisfied the foregoing condition are listed in Table 3.

## 5. Structure of the Neural Network Used in the Classification System

_{i}

_{, j}) with the dimensions N × M, and a load vector (bj) with the dimensions M × 1 was added. This operation in mathematical notation was expressed as follows:

^{−6}).

_{i}

_{,q}) with the dimensions M × K. Then, a load vector (bq) with the dimensions K × 1 was added. The next step involved activation by means of the activation function (φ), which was usually the Softmax function in classification issues.

_{q}) that a variable given to the input of the network belonged to a given class (label) at the output of the network:

_{i}

_{,q}—belonging index of the i-th sample to the q-th class;

_{i}

_{,q}—output value assigning the i-th sample (input) to the q-th class (output); probability and association by the network of the i-th input with the q-th output (class).

- Pump in working order;
- Pump at its end of life;
- Worn-out pump.

- 5 neurons in the input layer;
- 3 neurons in the classification layer;
- 12 neurons in the hidden layer.

#### Network Training—Learning Parameters

## 6. Analysis of Obtained Results and Final Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Simplified design diagram of an axial piston pump with a swashplate: 1—shaft, 2—rotor, 3—piston, 4—sliding shoe, 5—swashplate, 6—valve plate, 7—bearing [28].

**Figure 3.**Simplified diagram of the test bench: Pb—multi-piston pump being tested, Pw—booster pump, Zb—maximum valve, Zp—bypass valve, Zd—throttle valve, Zo—shut-off valve, Z1—non-return valve, Fs—low-pressure filter, Fw—filler filter, M—pressure gauge, n—rotometer, Q—flow meter, Ps—static pressure transducer, Pd—dynamic pressure transducer.

**Figure 4.**Significance ranking of features that satisfied the condition of the most significant applicability with minimal redundancy. Obtained from pump body vibration runs.

**Figure 5.**Significance ranking of the features that satisfied the condition of the greatest applicability with minimal redundancy. Obtained from the pressure waveforms (static and dynamic) in the discharge port of the pump.

**Figure 7.**Block structure of the network adopted for the classification of the wear state of the studied pump [40].

**Figure 8.**Waveforms of changes in accuracy and network error during training and validation process for the features of the received vibration signals.

**Figure 9.**Curves of changes in accuracy and network error during training and validation process for the features of the received pressure signals.

**Figure 10.**Matrix for classifying the wear state of the pump on the basis of features obtained from vibration signals.

**Figure 11.**Matrix for classifying the wear state of the pump based on features obtained from pressure signals.

No. | Expression: | Feature: | Description: |
---|---|---|---|

1 | $\stackrel{\u0332}{\mathrm{x}}=\frac{1}{\mathrm{n}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{x}}_{\mathrm{i}}$ | Mean | sum of all data divided by the number of data |

2 | $\mathsf{\delta}=\sqrt{{\mathsf{\delta}}^{2}}$ where ${\mathsf{\delta}}^{2}=\frac{1}{\mathrm{n}-1}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\left({\mathrm{x}}_{\mathrm{i}}-\stackrel{\u0332}{\mathrm{x}}\right)}^{2}$ | Standard deviation | square root of variance; the variance was estimated using a consistent and unbiased estimator |

3 | ${\mathrm{x}}_{\mathrm{RMS}}=\sqrt{\frac{1}{\mathrm{n}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{x}}_{\mathrm{i}}{}^{2}}$ | Root mean square | square root of the arithmetic mean of the data squared |

4 | ${\mathrm{x}}_{\mathrm{kurt}}=\frac{\frac{1}{\mathrm{n}}{\sum}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{i}}-\stackrel{\u0332}{\mathrm{x}}\right)}^{4}}{{\mathsf{\delta}}^{4}}$ | Kurtosis | measure of the shape of feature distribution |

5 | ${\mathrm{x}}_{\mathrm{skw}}=\frac{\frac{1}{\mathrm{n}}{\sum}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{i}}-\stackrel{\u0332}{\mathrm{x}}\right)}^{3}}{{\mathsf{\delta}}^{3}}$ | Skewness | defines the degree to which the distribution differs from the normal distribution |

6 | ${\mathrm{x}}_{\mathrm{sf}}=\frac{{\mathrm{x}}_{\mathrm{RMS}}}{\frac{1}{\mathrm{n}}{\sum}_{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{x}}_{\mathrm{i}}\right|}$ | Shape factor | root mean square of the signal divided by the mean value of the signal |

7 | ${\mathrm{x}}_{\mathrm{if}}=\frac{\mathrm{max}\left(\left|{\mathrm{x}}_{\mathrm{i}}\right|\right)}{\frac{1}{\mathrm{n}}{\sum}_{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{x}}_{\mathrm{i}}\right|}$ | Impulse Factor | maximum absolute value of the signal divided by the mean absolute value of the signal |

8 | ${\mathrm{x}}_{\mathrm{crest}}=\frac{\mathrm{max}\left(\left|{\mathrm{x}}_{\mathrm{i}}\right|\right)}{{\mathrm{x}}_{\mathrm{RMS}}}$ | Crest Factor | maximum absolute value in the data divided by the root mean square of the data |

9 | ${\mathrm{x}}_{\mathrm{clear}}=\frac{\mathrm{max}\left(\left|{\mathrm{x}}_{\mathrm{i}}\right|\right)}{{\left(\frac{1}{\mathrm{n}}{\sum}_{\mathrm{i}=1}^{\mathrm{n}}\sqrt{\left|{\mathrm{x}}_{\mathrm{i}}\right|}\right)}^{2}}$ | Clearance Factor | maximum absolute value of the signal divided by the square root of the signal amplitude |

10 | max PSD | Peak amplitude of PSD | maximum value of the power spectral density |

11 | max Freq. | Peak frequency of PSD | frequency of the maximum value of power spectral density |

xi—i-th measurement data point, n—total number of data in the measurement |

Feature | MIQ_{X} Coefficient Value |
---|---|

XT_PeakFreq | 0.37 |

YW_PeakAmp | 0.31 |

YT_CreastFactor | 0.05 |

XW_RMS | 0.015 |

YR_Skewness | 0.015 |

Feature | MIQ_{X} Coefficient Value |
---|---|

CD_PeakAmp | 0.33 |

CS_Skewness | 0.24 |

CS_Kurtosis | 0.07 |

CD_ImpulseFactor | 0.06 |

CS_PeakAmp | 0.05 |

Network Parameters | ||||
---|---|---|---|---|

No. | Name | Type | Number of Activated Variables | Variables Subject to Learning |

1 | WAR. WEJ. | Input layer | 5 | - |

2 | WAR. WEW. | Inner layer | 12 | 12 × 5 (weights), 12 × 1 (load) |

3 | ReLU | Activation | 12 | - |

4 | Batchnorm | Data normalization | 12 | 12 × 1 (offset), 12 × 1 (scaling) |

5 | WAR. WYJ. | Output layer | 3 | 3 × 12 (weights), 3 × 1 (load) |

6 | Softmax | Smoothing | 3 | - |

7 | KLASYFIKACJA | Classification | - | - |

No. | Parameter Name | Parameter Value |
---|---|---|

1 | Gradient Decay Factor | 0.95 |

2 | Squared Gradient Decay Factor | 0.99 |

3 | Epsilon | 1 × 10^{−8} |

4 | Initial Learn Rate | 1 × 10^{−3} |

5 | Learn Rate Schedule | ‘none’ |

6 | Learn Rate Drop Factor | 0.1 |

7 | Learn Rate Drop Period | 10 |

8 | L2Regularization | 1 × 10^{−3} |

9 | Gradient Threshold Method | ‘l2norm’ |

10 | Gradient Threshold | Inf. |

11 | Max Epochs | 500 |

12 | MiniBatch Size | 44 |

13 | Verbose | 1 |

14 | Verbose Frequency | 50 |

15 | Validation Data | 66 × 6 table |

16 | Validation Frequency | 20 |

17 | Validation Patience | Inf. |

18 | Shuffle | every epoch |

20 | Execution Environment | gpu |

Diagnostic Signal Type | Method of Determining Features | Accuracy | Error | ||
---|---|---|---|---|---|

Training (%) | Validation (%) | Training (-) | Validation (-) | ||

Vibrations | MRMR | 94 | 93.9 | 0.17 | 0.16 |

Static and dynamic pressure | MRMR | 98.7 | 96.7 | 0.07 | 0.11 |

Type of Diagnostic Signal | Method of Determining Features | Accuracy of acc Classifier (%) |
---|---|---|

Vibrations | MRMR | 95.5 |

Static and dynamic pressure | MRMR | 100 |

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## Share and Cite

**MDPI and ACS Style**

Konieczny, J.; Łatas, W.; Stojek, J.
Classification of Wear State for a Positive Displacement Pump Using Deep Machine Learning. *Energies* **2023**, *16*, 1408.
https://doi.org/10.3390/en16031408

**AMA Style**

Konieczny J, Łatas W, Stojek J.
Classification of Wear State for a Positive Displacement Pump Using Deep Machine Learning. *Energies*. 2023; 16(3):1408.
https://doi.org/10.3390/en16031408

**Chicago/Turabian Style**

Konieczny, Jarosław, Waldemar Łatas, and Jerzy Stojek.
2023. "Classification of Wear State for a Positive Displacement Pump Using Deep Machine Learning" *Energies* 16, no. 3: 1408.
https://doi.org/10.3390/en16031408